Quantum gate control · Chapter 02

Analytical structure first, learned corrections second

The controller does not construct every gate from scratch. It starts with a mathematical decomposition, represents nearby pulses in a small basis, and learns how to search that space using feedback.

A Lie-group decomposition becomes a pulse sequence

The KAK (Cartan) decomposition separates local rotations from the nonlocal part of a two-qubit gate. In this control model it is compiled into four local-control segments alternating with three exchange segments, each lasting 30 ns: 210 ns in total.

Identity

Cartan form of a two-qubit unitary

The $k_j$ are single-qubit special unitaries and the $c_j$ are real canonical coordinates, with a physically irrelevant global phase suppressed. $X,Y,Z$ denote Pauli matrices. This is the standard decomposition used as a reference, not a new theorem of this project.

\begin{gathered}U_\star=(k_1\otimes k_2)\,A(\mathbf c)\,(k_3\otimes k_4),\\ A(\mathbf c)=\exp[-i(c_1X\otimes X+c_2Y\otimes Y+c_3Z\otimes Z)].\end{gathered}

Current manuscript: analytical reference construction.

Model

The implemented control Hamiltonians

SQ segments supply local Z, X, and Y controls; TQ segments supply exchange control $g_r$ . Detuning persists through both. Local Z control is not active during exchange segments in this model. The subscript q identifies the qubit on which a Pauli operator acts.

\begin{aligned}H_{\mathrm{SQ},r}(t)&=\sum_{q=1}^2\big[(\delta_q+\alpha_{r,q}(t))Z_q\\&\hspace{48pt}+x_{r,q}(t)X_q+y_{r,q}(t)Y_q\big],\\ H_{\mathrm{TQ},r}(t)&=\delta_1Z_1+\delta_2Z_2+g_r(t)(X_1X_2+Y_1Y_2).\end{aligned}

Current manuscript: Physical Control Model.

Search in coefficients rather than arbitrary waveforms

Each scalar control is expanded in Walsh functions: orthogonal, piecewise-constant functions taking values +1 and −1, ordered by their number of sign changes. Changing a coefficient changes an entire structured part of the pulse, rather than an unrelated collection of time samples.

The main calibration setting uses two modes. One is constant; the other switches sign halfway through a segment. There are 27 scalar controls across the sequence, hence 54 coefficients. The controller proposes bounded residual changes around the analytical coefficients, subject to the control-amplitude constraints.

Definition

A 54-dimensional space of pulse commands

Time is measured from the start of each 30 ns segment. There are eight controls in each of the local Z/X/Y families and three exchange controls: $8+8+8+3=27$ . Two Walsh coefficients per scalar control give 54 commands; the physical pulse is reconstructed subject to amplitude constraints.

\begin{gathered}c_j(t)=\sum_{m=0}^{K-1}\beta_{m,j}w_m(t),\quad j=1,\ldots,27,\\ w_0(t)=1,\qquad w_1(t)=\begin{cases}+1,&0\le t<T_{\mathrm{seg}}/2,\\-1,&T_{\mathrm{seg}}/2\le t<T_{\mathrm{seg}},\end{cases}\\ \boldsymbol\beta=\boldsymbol\beta^0(U_\star)+\Delta\boldsymbol\beta\in\mathbb R^{54}\quad(K=2).\end{gathered}

Eight policies, one retained pulse

Eight policies, trained sequentially with TD3, an actor–critic reinforcement-learning algorithm, handle the two Walsh modes of four control families: local Z control, exchange strength, and the two transverse-drive components. Each action changes one coefficient group while leaving the others fixed. Every proposal reconstructs and evaluates the complete pulse.

A proposal is kept only when its fidelity improves on the retained pulse. The reward uses the change in a negative-log-infidelity score, including negative rewards for rejected proposals. One pass through all policies makes 56 proposals; repeated passes reuse the same frozen policies while retaining the improved pulse.

Proposition

Exact-feedback acceptance is monotone

For a fixed target and fixed noise realization, let $F(c)$ denote exact gate fidelity. Monotonicity follows from the acceptance rule; it is not a convergence or optimality guarantee for reinforcement learning.

c_{k+1}=\begin{cases}c_k^{\mathrm{prop}},&F(c_k^{\mathrm{prop}})>F(c_k),\\c_k,&\text{otherwise},\end{cases}\qquad F(c_{k+1})\ge F(c_k).

Proof / derivation

If accepted, the proposal has strictly larger fidelity. If rejected, the retained pulse is unchanged. These exhaust the cases. With noisy estimates, this argument applies only to the estimates—not necessarily to the true fidelity.

Elementary consequence of the implemented acceptance rule.

Quality feedback versus directional feedback

The fidelity-only controller receives gate quality, known pulse commands, and a short history of attempted corrections. It is not given the implemented channel. A newer variant adds selected local components of the residual error, projected along directions determined by the reference sequence.

These features connect the overall gate error to correction directions associated with particular segments. They are not separate measurements of error accumulated in each segment. In the current experiments they are computed from simulated channels; estimating them on hardware introduces a measurement requirement beyond scalar fidelity feedback.

Last updated September 27, 2026